# Syllabus Math 415

## Math 415. Applied Linear Algebra Instructor Syllabus

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• Standard text: Gilbert Strang, Linear Algebra and Its Applications, 4th Edition, Cengage Publishing
• Instructors can choose instead to use Bretscher, Linear Algebra with Applications, 4th Edition, Prentice Hall

Chapter 1. Matrices and Gaussian Elimination (6 hours)
1.3 An Example of Gaussian Elimination
1.2 The Geometry of Linear Equations
1.4 Matrix Notation and Matrix Multiplication
1.5 Triangular Factors and Row Exchanges
1.6 Inverses and Transposes
1.7 Special Matrices and Applications

Chapter 2. Vector Spaces (8 hours)
2.1 Vector Spaces and Subspaces
2.2 Solving Ax = 0 and Ax = b
2.3 Linear Independence, Basis, and Dimension
2.4 The Four Fundamental Subspaces
2.5 Graphs and Networks
2.6 Linear Transformations (supplemental notes available)

Chapter 3. Orthogonality (9 hours)
3.1 Orthogonal Vectors and Subspaces
3.2 Cosines and Projections onto Lines
3.3 Projections and Least Squares
3.4 Orthogonal Bases and Gram-Schmidt
5.5 Complex Matrices (first half – crash course in complex variables)
3.5 The Fast Fourier Transform (can be done as a handout – available)

Chapter 4. Determinants (3 hours)
4.2 Properties of the Determinant
4.3 Formulas for the Determinant
4.4 Applications of Determinants

Chapter 5. Eigenvalues and Eigenvectors (8 hours)
5.1 Introduction
5.2 Diagonalization of a Matrix
5.3 Difference Equations and Powers Ak
5.4 Differential Equations and eAt
5.5 Complex Matrices (second half – complex matrices)
5.6 Similarity Transformations (see supplemental notes on Linear Transformations)

Chapter 6. Positive Definite Matrices (4 hours)
6.1 Minima, Maxima, and Saddle Points
6.2 Tests for Positive Definiteness
6.3 Singular Value Decomposition

Exams, review and leeway (6 hours)